Conditions where random phase approximation becomes exact in the high-density limit

It is shown that, in d-dimensional systems, the vertex corrections beyond the random phase approximation (RPA) or GW approximation scales with the power d - beta - alpha of the Fermi momentum if the relation between Fermi energy and Fermi momentum is epsilon(f) similar to p(f)(beta) and the interacting potential possesses a momentum power law of similar to p(-alpha). The condition d - beta - alpha < 0 specifies systems where RPA is exact in the high-density limit. The one-dimensional structure factor is found to be the interaction-free one in the high-density limit for contact interaction. A cancellation of RPA and vertex corrections render this result valid up to second order in contact interaction. For finite-range potentials of cylindrical wires a large-scale cancellation appears and is found to be independent of the width parameter of the wire. The proposed high-density expansion agrees with the quantum Monte Carlo simulations.